Since investors like to increase their expected wealth and like to avoid risk or uncertainty, it is possible to imagine different combinations of expected gain and risk which are valued equally by an investor. That is, an investor will be willing to assume greater risk, if he achieves greater expected wealth.
The individual investor is now conceptually prepared to select the optimum portfolio from those constituting the efficient set. The optimum portfolio (i.e., the one which maximizes expected utility) is the one at the point of tangency between the efficient frontier and an indifference curve. In images it can be seen that the investor can do no better than choose the portfolio at point A on the efficient frontier, since no other portfolio is on as high an indifference. Another escape is to say that concavity does not necessarily imply that the relationship is quadratic and that other equations can preserve the concavity without ever implying a maximum value from which utility will decline as wealth increases. The difficulty with these other curves is that efficiency in terms of the mean and variance of a portfolio does not necessarily imply maximization of expected utility. Markowitz has shown, however, that many utility functions can be reasonably approximated by the quadratic.
A different line of criticism has been advanced by Arditti and others. They argue that investors may be interested in characteristics of distributions of rates of return additional to the mean and variance. In particular, they argue that skewness may be of importance. That is, if the rates of return on the portfolios have the same mean and variance, but different skewness, investors may prefer the distribution which is more skewed to the right. One is not excused from reaching tentative conclusions simply because the theoretical development of a field is still rudimentary. A conclusion which is consistent with much that has been observed in the real world and which is satisfying theoretically is the one with which we started: namely, that portfolios which are efficient in terms of their means and variances necessarily maximize expected utility which can be represented by a quadratic equation. Markowitz, perhaps, does the best job of showing that his efficient portfolios are very close to optimum or come very close to maximizing expected utility, even if things other than the mean and variance of the distributions of returns make a difference to or affect the expected utility of inves tors. Even if the investor is concerned about the magnitude of the expected loss, the maximum expected loss, the probability of a loss, or other attributes of the distribution, the portfolios selected according to those criteria will be very similar to portfolios selected according to their means and variances.